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Position a rack in the center of the oven and heat the oven to 375 degrees F.

In a medium bowl, stir together the graham cracker crumbs and 3 Tbs. granulated sugar. Mix in the melted butter until the crumbs are evenly moist and clump together slightly. Transfer the mixture to a 9-inch springform pan and press evenly onto the bottom and about 2 inches up the sides of the pan (to press, use plastic wrap or a flat-bottom measuring cup). Bake until the crust is fragrant and slightly darkened, 9 to 12 minutes. Let the pan cool on a rack. Lower the oven temperature to 300 degrees F.

In a stand mixer fitted with the paddle attachment, beat the cream cheese, banana puree, flour, and a pinch of table salt on medium speed, scraping down the sides of the bowl and the paddle frequently, until very smooth and fluffy, about 5 minutes. Make sure the cheese has no lumps. Add the 1-1/4 cups granulated sugar and continue beating until well blended and smooth.

Add the chocolate chips and vanilla, and beat until blended, about 30 seconds. Add the eggs one at a time, beating just until blended. (Don't overbeat once the eggs have been added or the cheesecake will puff too much and crack as it cools.) Pour the filling into the cooled crust and smooth the top.

Bake at 300 degrees F until the center jiggles like Jell-O when nudged, 55 to 65 minutes. The cake will be slightly puffed around the edges, and the center will still look moist. Set on a rack and cool completely. Cover and refrigerate until well chilled, at least 8 hours and up to 3 days. The cake can also be frozen at this point for up to 1 month. (To freeze, put the unmolded, cooled cake on a rimmed baking sheet in the freezer, uncovered, until the top is cold and firm; then wrap it in two layers of plastic and one layer of foil. Thaw overnight in the refrigerator.)

Unclasp and remove the side of the springform pan and run a long, thin metal spatula under the bottom crust. Carefully slide the cake onto a flat serving plate. Arrange the halved strawberries on top of the cake. If you want to glaze the fruit, heat the jam in a small saucepan with 1 Tbs. water, stirring frequently, until melted and smooth; strain. Brush the strawberries with the melted jam mixture.

Garnish with chocolate shards and serve immediately. To cut, run a thin knife under hot water, wipe it dry, and cut the cake into slices, heating and wiping the knife after every slice.